1

Mathematical Models of Mathematical Models of RunningRunning

Dan WhittDan Whitt

Wednesday September 24, 2008Wednesday September 24, 2008UMS TalkUMS Talk

2

ReferencesReferencesAlvarez-Ramirez, Jose. An Improved Peronnet-Thebault mathematical model of human Running

performance, Euro. J. Appl. Physiology, 86,517-525, 2002.

Behncke, Horst. Optimization models for the force and energy in competitive running. J. Math. Biol. 35,375-390, 1997.

Furusawa K., A.V. Hill, and J.L. Parkinson. The Energy Used in Sprint Running. Proceedings of the Royal Society of London, Series B, Containing Papers of a Biological Character, Vol. 102, No. 713, 43-50, 1927.

Hill, A.V. The Air-Resistance to a Runner. Proceedings of the Royal Society of London, Series B, Containing Papers of a Biological Character, Vol. 102, No. 718, 380-385, 1928.

Keller, Joseph B. A Theory of Competitive Running, Physics Today, 26, No. 9, 42-47, 1973.

Keller, Joseph B. Optimal Velocity in a Race, Keller, Joseph B. Optimal Velocity in a Race, Am. Math. MonthlyAm. Math. Monthly 8181, 474, 474--480, 1974480, 1974.

Kirk, Donald E. Optimal Control Theory: An Introduction. Dover Publications: New York, 2004.

Kram, Rodger and C. Richard Taylor. Energetics of running: A new perspective. Nature, 346, July 1990.

Pritchard, W.G. Mathematical Models of Running. SIAM Review, 35, No. 3, 359-279, 1993.

Woodside, William. The Optimal Strategy for Running a Race (A Mathematical Model for World Records from 50m to 275km), Mathl. Comput. Modeling, 15, No. 10, 1-12, 1991.

3

The Pioneer: A.V. HillThe Pioneer: A.V. Hill

Nobel Prize, 1922.Hill received the 1922 Nobel Prize for Physiology or Medicine "for his discovery relating to the production of heat in muscle."

Picture found in: Bassett Jr., David R. Scientific Contributions of A.V. Hill: exercise physiology pioneer. J Appl Physiology. 93: 1567-1582, 2002.

4

Why investigate athletics?Why investigate athletics?

The complaint has been made to me "Why investigate athletics, whThe complaint has been made to me "Why investigate athletics, why not y not study the processes of industry or of disease?" The answer is twstudy the processes of industry or of disease?" The answer is twofold. (1) The ofold. (1) The processes of athletics are processes of athletics are simple and measurablesimple and measurable and carried out to a and carried out to a constant degree, namely to the utmost of a man's powers: those oconstant degree, namely to the utmost of a man's powers: those of industry f industry are not; and (2) athletes themselves, being in a state of healthare not; and (2) athletes themselves, being in a state of health and dynamic and dynamic equilibrium, can be equilibrium, can be experimented on without dangerexperimented on without danger and can repeat their and can repeat their performances exactly again and again. performances exactly again and again.

I might perhaps state a third reason and say, as I said in PhilaI might perhaps state a third reason and say, as I said in Philadelphia, that the delphia, that the study of athletes and athletics is "study of athletes and athletics is "amusingamusing": certainly to us and sometimes I ": certainly to us and sometimes I hope to them. Which leads to a fourth reason, perhaps the most ihope to them. Which leads to a fourth reason, perhaps the most important of mportant of all: that being "amusing" it may help to bring new and enthusiasall: that being "amusing" it may help to bring new and enthusiastic recruits to tic recruits to the study of physiology, which needs every one of them, especialthe study of physiology, which needs every one of them, especially if they be ly if they be chemists. chemists.

Quotations found in: Bassett Jr., David R. Scientific Contributions of A.V. Hill: exercise physiology pioneer. J Appl Physiology. 93: 1567-1582, 2002.

5

ParaphrasedParaphrased

Athletics is simple, measurableAthletics is simple, measurable

Athletes are good experimental subjectsAthletes are good experimental subjects

ItIt’’s cool!s cool!

6

A.V. Hill ModelA.V. Hill Model

Based on NewtonBased on Newton’’s Laws and s Laws and ThermodynamicsThermodynamics

The ideal Human Engine is 38% efficientThe ideal Human Engine is 38% efficient

Not much mathematics. A lot of physiology.Not much mathematics. A lot of physiology.

Hill, A.V. The AirHill, A.V. The Air--Resistance to a Runner. Resistance to a Runner. Proceedings of the Royal Society of London, Series B, Proceedings of the Royal Society of London, Series B, Containing Papers of a Biological Character, Containing Papers of a Biological Character, Vol. 102, No. 718, 380Vol. 102, No. 718, 380--385, 1928.385, 1928.

7

Joseph B. KellerJoseph B. Keller

400+ publications!400+ publications!Developed the Developed the Geometrical Theory of Geometrical Theory of DiffractionDiffractionDeveloped the EBK Developed the EBK (Einstein(Einstein--BrillouinBrillouin--Keller) method of Keller) method of quantizationquantization

Biographical info and picture from: http://Biographical info and picture from: http://math.stanford.edu/~kellermath.stanford.edu/~keller//

8

The Keller ModelThe Keller Model

Built primarily upon the physical foundations of the Hill Model

1.1. Conservation of Linear MomentumConservation of Linear Momentum

2.2. Conservation of EnergyConservation of Energy

3.3. Oxygen consumption yields energyOxygen consumption yields energy

4. Cast as an Optimal Control Problem

Keller, Joseph B. A Theory of Competitive Running, Keller, Joseph B. A Theory of Competitive Running, Physics TodayPhysics Today, , 2626, No. 9, 42, No. 9, 42--47, 1973.47, 1973.

Keller, Joseph B. Optimal Velocity in a Race, Keller, Joseph B. Optimal Velocity in a Race, Am. Math. MonthlyAm. Math. Monthly 8181, 474, 474--480, 1974480, 1974.

9

KellerKeller’’s Problems Problem

““We wish to determine how a runner should vary We wish to determine how a runner should vary his speed his speed v(t)v(t) during a race of distance during a race of distance DD in in order to run it in the shortest time.order to run it in the shortest time.””

10

VariablesVariables

v(t) = velocity

f(t) = propulsive force per unit mass

E(t) = available energy (oxygen) in the muscles per unit mass

T = total time

D = total distance

11

Physiological ConstantsPhysiological Constants

F = maximum possible value of f(t)F = maximum possible value of f(t)

EE00 = E= E(0)(0), initial available energy (oxygen), initial available energy (oxygen)

σσ = rate at which energy (oxygen) is supplied= rate at which energy (oxygen) is supplied

ττ = a constant related to the resistive force per unit = a constant related to the resistive force per unit massmass

12

Six Defining EquationsSix Defining Equations

0)0(

)(

)(0

=

=+

= ∫

v

tfτv

dtdv

dttvDT

0)(

)()(

)(

≥

−=

≤

tE

tvtfdtdE

Ftf

σ

4.1.

2. 5.

6.3.

13

Mathematical ProblemMathematical Problem

Given T, and the four physiological constants,

Find the three functions v(t), f(t) and E(t) satisfying the six equations such that

is maximized.∫=T

dttvD0

)(

14

SimplificationSimplification

Express Express ff((tt)) and and EE((tt) in terms of ) in terms of vv((tt)) by by substitution within the six defining substitution within the six defining equations. equations.

15

Fvdtdv

≤+⇒τ

)(tfvdtdv

=+τ

Ftf ≤)(and

Eqn. 2 Eqn. 4 Eqn. 7

⎟⎠⎞

⎜⎝⎛ +−=⇒−=

τσσ )()()()( tv

dtdvtv

dtdEtvtf

dtdE

Eqn. 5

Then integrate the last expression to obtain:

0)(1)()(0

22

0 ≥−−+= ∫t

dssvtvtEtEττ

σ

Eqn. 8

16

Reformulated ProblemReformulated Problem

∫=T

dttvD0

)(Maximize such that

,Fvdtdv

≤+τ

,0)0( =v1. 2.

0)(1)(

0

22

0 ≥−−+ ∫t

dssvtvtEττ

σ3.

17

First: The SolutionFirst: The Solution

Then: Some ComputationThen: Some Computation

18

Solution: 2 CasesSolution: 2 Cases

The Race is a short sprint or The Race is a short sprint or ““dash,dash,”” with with typical parameters, meters. typical parameters, meters.

The Race is longer distance.The Race is longer distance.

291≤D

19

Case 1: The DashCase 1: The Dash

Ftf =)( for all t.

)1()( /ττ teFtv −−=

⎟⎠⎞

⎜⎝⎛ −+=⇒ − 1/2 τ

ττ TeTFD cTT ≤≤0

20

Case 1: The Graph of Case 1: The Graph of vv((tt))

)1()( /ττ teFtv −−=

21

Case 2: The Longer DistancesCase 2: The Longer Distances

cc DDTT ≥⇔≥

3 Phases

1.Acceleration:

2.Cruising: v(t) = constant

3.Deceleration:

)1()( /ττ teFtv −−= 10 tt ≤≤

21 ttt ≤≤

Ttt ≤≤2

( ) 2/1/)(22

2 2])([)( τστστ ttetvtv −−+=

22

Case 2: The Graph of Case 2: The Graph of vv((tt))

( )τστστ /)(22

2 2])([)( ttetvtv −−+=

Ttt ≤≤2

),1()( /ττ teFtv −−= 10 tt ≤≤

,/)( λτ=tv 21 ttt ≤≤

( ) 2/1/)(22

2 2])([)( τστστ ttetvtv −−+=

Ttt ≤≤2

23

The DetailsThe Details……

Comic Found online: http://www.math.kent.edu/~sather/PHOTOS/math03.gif

24

Calculus of VariationsCalculus of Variations

The calculus of The calculus of functionalsfunctionals..

Functional = Functional = ““function of a functionfunction of a function””

The goal is usually associated with finding a The goal is usually associated with finding a maximizing or minimizing function of a given maximizing or minimizing function of a given functional.functional.

See Kirk and WikipediaSee Kirk and Wikipedia

25

More Precise DefinitionsMore Precise Definitions

26

One Key Step in Case 2 One Key Step in Case 2

After we determine an expression for After we determine an expression for D D > > DDcc, we must choose , we must choose vv to maximize to maximize DD subject subject to the constraint to the constraint EE((tt22) = ) = 00..

To do this, we consider the functional To do this, we consider the functional DD + + λλEE((tt22)/2 where )/2 where λλ is a Lagrange multiplier.is a Lagrange multiplier.

We then compute the first variation of this We then compute the first variation of this functional and set it equal to zero.functional and set it equal to zero.

27

ComputationComputation……

Image found at: http://home.uchicago.edu/~slapan/Links/images/Math.gif

28

OK, you can wake up now.OK, you can wake up now.

29

WhatWhat’’s Missing?s Missing?

Accurate for races 400m and less.Accurate for races 400m and less.

What really happens in the races longer than 400m? What really happens in the races longer than 400m?

Even in Even in ““solosolo”” world record attempts the races are world record attempts the races are run with a run with a ““negative split.negative split.””

There is typically an acceleration in the later part of There is typically an acceleration in the later part of the race, either quickly and forcefully to defeat the race, either quickly and forcefully to defeat competitors, or gradually over the final third of the competitors, or gradually over the final third of the race to expend all remaining energy more evenly.race to expend all remaining energy more evenly.

30

How can we revise the model?How can we revise the model?

Add variables to improve the accuracy of the Add variables to improve the accuracy of the fatigue factors.fatigue factors.

Use a 3 compartment hydraulic model to Use a 3 compartment hydraulic model to more accurately represent our knowledge of more accurately represent our knowledge of physiology physiology –– see see BehnckeBehncke

Is there another way?Is there another way?

31

UncertaintyUncertainty

When D > DWhen D > Dcc, there exists some uncertainty , there exists some uncertainty about what constant rate to attempt between about what constant rate to attempt between time ttime t11 and tand t22. .

The athlete does not know the values of EThe athlete does not know the values of E00

and and σσ and only learns once the race has and only learns once the race has begun based upon feelings.begun based upon feelings.

32

New Problem, New SolutionNew Problem, New Solution

Problem: Determine the best pacing strategy given Problem: Determine the best pacing strategy given your objective. your objective.

Put a probability distribution on Put a probability distribution on σσ and and EE00. .

Then choosing KellerThen choosing Keller’’s pace will result in a 50% s pace will result in a 50% failure rate. failure rate.

Perhaps you want a greater chance of finishing Perhaps you want a greater chance of finishing before running out of energy, be conservative.before running out of energy, be conservative.

Or perhaps you want to risk it, go out hard, test Or perhaps you want to risk it, go out hard, test your limits.your limits.